Co-prime array coherent source DOA estimation method for low signal-to-noise ratio and small sample number

Abstract

The invention discloses a co-prime array coherent source DOA (Direction of Arrival) estimation method aiming at a low signal-to-noise ratio and a small sample number, and relates to a subsurface buoy for ocean observation and a target detection processing method of a mobile observation platform. Specifically, coprime array output is converted into virtual array output, then an interpolation covariance matrix is obtained, then a covariance matrix is constructed by using a non-zero column, and an augmented covariance matrix is recovered through atom norm optimization. And finally, realizing coprime array coherent source DOA estimation under the condition of low signal-to-noise ratio and small sample number by using a coprime array PM algorithm. Compared with a traditional method, the method has the advantages that effective DOA estimation of the coherent source under the co-prime array condition is achieved, the DOA estimation precision is improved in the complex environment with the low signal-to-noise ratio and the small sample number, and the method has important application value in actual engineering.

Description

Technical Field

[0001] The present invention relates to a target detection and processing method for a buoy and a mobile observation platform for ocean observation. The method is a DOA estimation method for coprime array coherent sources under low signal-to-noise ratio and small sample number. Specifically, the method converts the coprime array output into a virtual array output to obtain an interpolation covariance matrix, then constructs the covariance matrix using non-zero columns, and restores the augmented covariance matrix through atomic norm optimization. Finally, the method uses a coprime array PM algorithm to realize DOA estimation of coprime array coherent sources under low signal-to-noise ratio and small sample number. Background Art

[0002] In the past decade, coprime arrays have attracted widespread attention in the field of DOA estimation because they can be expanded into virtual uniform linear arrays with higher degrees of freedom and suffer less coupling effects between array elements. In the study of coprime arrays, many methods for DOA estimation have been proposed. However, most methods are only for signals under ideal conditions. Due to the complex underwater acoustic environment, the passive sonar signals of weak targets in the deep sea are greatly affected by ocean noise. When the signal is overwhelmed by noise, in the case of low signal-to-noise ratio, traditional algorithms fail and more robust estimation methods are needed. In the case of small snapshot numbers, the covariance matrix estimation is inaccurate and the performance of subspace algorithms drops sharply. Therefore, the study of DOA estimation methods under low signal-to-noise ratio and small sample number is an important part of array signal processing. In order to improve the target detection accuracy of coprime arrays in harsh environments, it can be effectively applied to real life.

[0003] When coherence exists between signal sources, traditional coprime array methods face severe performance degradation. Traditional coherent signal DOA estimation methods, such as the forward and backward MUSIC algorithm, suffer from aperture loss. The ESPRIT algorithm exploits the rotational invariance of the array, eliminating the need for eigenvalue decomposition and offering high computational efficiency. However, it is only applicable to uniform arrays. Therefore, DOA estimation of coherent sources in coprime arrays remains a hot topic of research and is closely related to social development and stability.

[0004] This patent proposes a method for DOA estimation of coherent sources in coprime arrays under low signal-to-noise ratio and small sample size conditions. This method converts the coprime array outputs into virtual array outputs, obtains the interpolated covariance matrix, constructs the covariance matrix using nonzero columns, optimizes the atomic norm to restore the augmented covariance matrix, and uses the coprime array PM algorithm to achieve DOA estimation of coherent sources under low signal-to-noise ratio and small sample size conditions. This method effectively estimates DOA of coprime array coherent sources and has significant results in complex environments with low signal-to-noise ratio and small sample size, making it of great application value in practical engineering. Summary of the Invention

[0005] The actual ocean acoustic environment is very complex and contains many non-ideal factors, which results in a low signal-to-noise ratio (SNR) and a low number of snapshots in the estimated environment. To improve the target detection accuracy of coprime arrays and solve the problem of low DOA estimation accuracy under low SNR and small sample sizes, the present invention proposes a DOA estimation method for coprime array coherent sources under low SNR and small sample sizes. The method converts the coprime array output into a virtual array output, obtains an interpolated covariance matrix, constructs the covariance matrix using non-zero columns, optimizes the atomic norm to restore the augmented covariance matrix, and uses the coprime array PM algorithm to implement a DOA estimation method for coprime array coherent sources under low SNR and small sample sizes.

[0006] The present invention provides a method for estimating the DOA of coherent sources in a coprime array under low signal-to-noise ratio and small sample number, characterized in that: the coprime sensor array is constructed as follows: the number of sensors in uniform subarray 1 is N, the sensor spacing is Md, the number of sensors in uniform subarray 2 is M, the sensor spacing is Nd, d=2 / λ, λ is the wavelength, the coprime array is composed of a nested uniform subarray 1 and a uniform subarray 2, the first array elements of the two subarrays are overlapped, the coprime array position set is expressed as S={nM|0≤n≤N-1}∪{mN|0≤m≤M-1}, the number of sensors is Z=M+N-1, and the method for estimating the DOA of coherent sources in a coprime array under low signal-to-noise ratio and small sample number comprises the following steps:

[0007] Step 1: K coherent far-field narrowband signals are incident on the coprime array. The coprime array receives data that can be expressed as X(t) = [x1(t), x2(t), ..., x M+N-1 (t)] T , perform Hilbert transform on these data, and get the complex signal form of the array receiving signal:

[0008] Step 2: Use the coprime array interpolation to obtain the coprime array virtual array structure V = {l, 0≤l≤max(S)}. The received data of the coprime array virtual array can be constructed as Where l∈V\S means l belongs to V but not to S. The received data of the coprime virtual array can be expressed as X V (t) = [x V1 (t),x V2 (t),…,x Vmax(S) (t)] T , perform Hilbert transform on these data, and get the complex signal form of the virtual array receiving signal:

[0009] Step 3: Use the complex signal of the virtual array Find the covariance matrix “(·) H” is the conjugate transpose, where T is the number of snapshots in practical applications;

[0010] Step 4. Select The non-zero columns in , get the new matrix where J = [e1, e2,…, e |S| ]∈R |V|×|S| is a selection matrix, e i is except the first (l i +1) positions except for the 1 column vector,

[0011] Step 5: Generate matrix H, H is The zero matrix of the same dimension is the corresponding The elements at the positions corresponding to the zero items are set to 0, and the elements at the other positions are set to 1;

[0012] Step 6: To fill the holes, the matrix filling problem can be implemented using the following atomic norm optimization model: by solving the corresponding semi-definite programming problem To optimize and fill the holes, use After optimization and in is a complex matrix, So u∈C |V|×1 is the Hermitian, Toeplitz, or positive semidefinite matrix of the first column vector, where Y∈C |S|×|S| “tr(·)” represents the trace of the matrix, and “(·) H " represents the conjugate transpose of the matrix, min represents minimization, μ is the regularization parameter, represents the Hadamard product, “|| || F " represents the Frobenius norm of the matrix, "≥" represents semi-positive definiteness, and semi-positive definite programming problems can be solved using the CVX convex optimization toolbox;

[0013] Step 7: Divide into blocks and get where D∈C |V|×|K| , E∈C |V|×|V-K| , K is the number of incident signals;

[0014] Step 8. Calculate D and E in step 7 to obtain The minimum solution is P=(D H D) -1 D H E;

[0015] Step 9: The projection operator introduced into the noise subspace can be orthogonalized to obtain the matrix Q = [P H -I L-K ], where I represents the unit matrix, I∈R (L-K)×(L-K) , L is the length of the virtual array structure V;

[0016] Step 10: According to the formula Calculate the spatial spectrum, where a(θ) represents the array steering vector of the incident signal. Perform a peak search on the spatial spectrum, and the angle corresponding to its extreme value is the estimated value of the incident signal angle.

[0017] The present invention adopts the above technical solution to achieve the following beneficial effects:

[0018] (1) The present invention realizes DOA estimation of coherent sources under coprime array conditions by converting coprime array outputs into virtual array outputs, obtaining an interpolated covariance matrix, constructing a covariance matrix using non-zero columns, and restoring the augmented covariance matrix through atomic norm optimization.

[0019] (2) The present invention introduces the PM algorithm into the DOA estimation of coprime array coherent sources and combines it with atomic norm optimization to effectively improve the accuracy of DOA estimation under the conditions of low signal-to-noise ratio and small sample number; BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 This is a model diagram of the array signal processing method of this patent;

[0021] Figure 2 This is the DOA estimation result diagram of the patented signal processing method;

[0022] Figure 3 This is the relationship curve between the probability of successful resolution and the signal-to-noise ratio of the patented signal processing method;

[0023] Figure 4 This is the relationship curve between the root mean square error and the signal-to-noise ratio of the patented signal processing method;

[0024] Figure 5 This is the relationship curve between the probability of successful resolution and the number of snapshots of the patented signal processing method;

[0025] Figure 6 This is the relationship curve between the root mean square error and the number of snapshots of the patented signal processing method. DETAILED DESCRIPTION

[0026] The present invention will now be further described with reference to the embodiments and accompanying drawings:

[0027] The first embodiment: Figure 1The coprime array model of the present invention is given. First, we construct a 7-element coprime sensor array. The number of sensors in uniform subarray 1 is 3, and the sensor spacing is 5d. The number of sensors in uniform subarray 2 is 5, and the sensor spacing is 3d. d = 2 / λ, where λ is the wavelength. The coprime array is composed of a nested uniform subarray 1 and a nested uniform subarray 2. The first elements of the two subarrays are overlapped. The coprime array position set is represented by S = {5n|0≤n≤2}∪{3m|0≤m≤4}. The number of sensors is Z = 7. A coprime array coherent source DOA estimation method for low signal-to-noise ratio and small sample size includes the following steps:

[0028] Step 1: Two coherent far-field narrowband signals are incident on a coprime array. The received data of the coprime array can be expressed as X(t) = [x1(t), x2(t), …, x7(t)] T , perform Hilbert transform on these data, and get the complex signal form of the array receiving signal:

[0029] Step 2: Use the coprime array interpolation to obtain the coprime array virtual array structure V = {l, 0≤l≤12}. The received data of the coprime array virtual array can be constructed as Where l∈V\S means l belongs to V but not to S. The received data of the coprime virtual array can be expressed as X V (t) = [x V1 (t),x V2 (t),…,x V13 (t)] T , perform Hilbert transform on these data, and get the complex signal form of the virtual array receiving signal:

[0030] Step 3: Use the complex signal of the virtual array Find the covariance matrix “(·) H ” is the conjugate transpose, where T is the number of snapshots in practical applications;

[0031] Step 4. Select The non-zero columns in , get the new matrix where J = [e1,e2,...,e |13| ]∈R |13|×|7| is a selection matrix, e i is except the first (l i +1) positions except for the 1 column vector,

[0032] Step 5: Generate matrix H, H is The zero matrix of the same dimension is the corresponding The elements at the positions corresponding to the zero items are set to 0, and the elements at the other positions are set to 1;

[0033] Step 6: To fill the holes, the matrix filling problem can be implemented using the following atomic norm optimization model: by solving the corresponding semi-definite programming problem To optimize and fill the holes, use After optimization and in is a complex matrix, So u∈C |13|×1 is the Hermitian, Toeplitz, or positive semidefinite matrix of the first column vector, where Y∈C |7|×|7| “tr(·)” represents the trace of the matrix, and “(·) H " represents the conjugate transpose of the matrix, min represents minimization, μ is the regularization parameter, represents the Hadamard product, “|| || F " represents the Frobenius norm of the matrix, "≥" represents semi-positive definiteness, and semi-positive definite programming problems can be solved using the CVX convex optimization toolbox;

[0034] Step 7: Divide into blocks and get where D∈C |13|×|2| , E∈C |13|×|13-2| , K is the number of incident signals;

[0035] Step 8. Calculate D and E in step 7 to obtain The minimum solution is P=(D H D) -1 D H E;

[0036] Step 9: The projection operator introduced into the noise subspace can be orthogonalized to obtain the matrix Q = [P H -I 13-2 ], where I represents the unit matrix, I∈R (13-2)×(13-2) , L is the length of the virtual array structure V;

[0037] Step 10: According to the formula Calculate the spatial spectrum, where a(θ) represents the array steering vector of the incident signal. Perform a peak search on the spatial spectrum, and the angle corresponding to its extreme value is the estimated value of the incident signal angle.

[0038] analyze Figure 2It can be seen that the incident angles of the two coherent signals are -20° and 20° respectively. When the signal-to-noise ratio is 5dB and the number of snapshots is 50, the angle estimated by the patented method is basically consistent with the true angle. Therefore, the patented method can effectively estimate DOA under low signal-to-noise ratio and small sample number.

[0039] Second embodiment: The relationship between the root mean square error (RMSE) and the signal-to-noise ratio and the relationship between the probability of successful resolution and the signal-to-noise ratio of the signal processing method of this patent are studied respectively, and the effect diagram is as follows: Figure 3 、 Figure 4 The application conditions of the algorithm in the present invention are as follows:

[0040] We used a 7-element coprime sensor array. Uniform subarray 1 had 3 sensors with a sensor spacing of 5d, and uniform subarray 2 had 5 sensors with a sensor spacing of 3d, where d = 2 / λ, and λ is the wavelength. The coprime array was composed of a nested uniform subarray 1 and a uniform subarray 2. The first elements of the two subarrays overlapped. The coprime array position set was represented as S = {5n|0≤n≤2}∪{3m|0≤m≤4}, and the number of sensors was Z = 7. Two coherent signals were incident on the coprime array from the directions of θ1 = -20° and θ2 = 20°, respectively. The number of snapshots was T = 150, and the signal-to-noise ratio was varied from -5dB to 25dB in steps of 5dB. 200 independent Monte Carlo experiments were performed, and simulations were performed using the MATLAB software system.

[0041] analyze Figure 3 It can be seen that when the number of snapshots T = 150, as the signal-to-noise ratio increases, the success probability of the patented method and the CCANM method increases. The success probability of the patented method reaches 100% when the signal-to-noise ratio is 0dB, and the success probability of the CCANM method is 99%. The success probability of the CCANM method reaches 100% when the signal-to-noise ratio is 5dB. This shows that the patented method can successfully distinguish the true angle, and the success probability is higher than other methods at low signal-to-noise ratios.

[0042] analyze Figure 4It can be seen that when the snapshot number T = 150, as the signal-to-noise ratio increases, the root mean square error of the two methods decreases. When the signal-to-noise ratio is -5dB, the root mean square error of the patented method is about 0.678, and the root mean square error of the CCANM method is about 0.789; when the signal-to-noise ratio is 0dB, the root mean square error of the patented method is about 0.528, and the root mean square error of the CCANM method is about 0.578; when the signal-to-noise ratio is 5dB, the root mean square error of the patented method is about 0.369, and the root mean square error of the CCANM method is about 0.397; in a high signal-to-noise ratio environment with a signal-to-noise ratio ranging from 10dB to 25dB, the root mean square errors of the two methods are basically the same. This shows that the performance of the patented method in DOA estimation is better than that of the CCANM method in a low signal-to-noise ratio environment, and its performance is basically the same as that of the CCANM method in a high signal-to-noise ratio environment.

[0043] The third embodiment: The relationship between the root mean square error (RMSE) and the number of snapshots and the relationship between the probability of successful resolution and the number of snapshots of the signal processing method of the patent are studied respectively, and the effect diagram is shown as follows: Figure 5 、 Figure 6 The application conditions of the algorithm in the present invention are as follows:

[0044] We use a 7-element coprime sensor array. Uniform subarray 1 has 3 sensors with a sensor spacing of 5d, and uniform subarray 2 has 5 sensors with a sensor spacing of 3d, where d = 2 / λ, and λ is the wavelength. The coprime array is composed of a nested uniform subarray 1 and a uniform subarray 2. The first elements of the two subarrays are placed overlapping. The coprime array position set is represented as S = {5n|0≤n≤2}∪{3m|0≤m≤4}, and the number of sensors is Z = 7. Two coherent signals are incident on the coprime array from the directions of θ1 = -20° and θ2 = 20°, respectively. The signal-to-noise ratio (SNR) is 10dB. The number of snapshots is changed from 50 to 400 in steps of 50. 200 independent Monte Carlo experiments are performed, and simulations are performed using the MATLAB software system.

[0045] analyze Figure 5 It can be seen that when the signal-to-noise ratio (SNR) is 0, the success probability of both the patented method and the CCANM method increases with the increase in the number of snapshots. When the number of snapshots is 50, the success probability of the patented method is 94.5%, and the success probability of the CCANM method is 88%. When the number of snapshots is 100, the success probability of the patented method is 99.5%, and the success probability of the CCANM method is 96%. When the number of snapshots is 150, the success probability of the patented method reaches 100%, while the success probability of the CCANM method is 99.5%. This shows that the patented method can successfully distinguish the true angle, and the success probability is higher than that of other methods at low snapshot numbers.

[0046] analyze Figure 6It can be seen that when the signal-to-noise ratio (snr) is 0, the root mean square error (RMS) of both methods decreases as the number of snapshots increases. When the number of snapshots is 50, the RMS error of the proposed method is approximately 0.601, while that of the CCANM method is approximately 0.634. When the number of snapshots is 100, the RMS error of the proposed method is approximately 0.537, while that of the CCANM method is approximately 0.592. When the number of snapshots is 200, the RMS error of the proposed method is approximately 0.503, while that of the CCANM method is approximately 0.557. This shows that the performance of the proposed method in DOA estimation is better than that of the CCANM method when the number of snapshots is small.

[0047] Although some embodiments of the present invention have been shown and described, it will be apparent to those skilled in the art that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and all resulting solutions fall within the scope of protection of the present invention.

Claims

1. A method for DOA estimation of coherent sources in a coprime array under low signal-to-noise ratio and small sample size conditions. The coprime sensor array is constructed as follows: uniform subarray 1 has N sensors and a sensor spacing of Md; uniform subarray 2 has M sensors and a sensor spacing of Nd, where d = 2 / λ, and λ is the wavelength. The coprime array is constructed by nesting uniform subarray 1 and uniform subarray 2, with the first elements of the two subarrays overlapping. The coprime array position set is represented by S = {nM|0≤n≤N-1}∪{mN|0≤m≤M-1}, and the number of sensors is Z = M+N-1. The method is characterized by: A method for estimating the DOA of coprime array coherent sources under low signal-to-noise ratio and small sample number includes the following steps: Step 1: K coherent far-field narrowband signals are incident on the coprime array. The coprime array receives data that can be expressed as X(t) = [x1(t), x2(t), ..., x M+N-1 (t)] T , perform Hilbert transform on these data, and get the complex signal form of the array receiving signal: Step 2: Use the coprime array interpolation to obtain the coprime array virtual array structure V = {l, 0≤l≤max(S)}. The received data of the coprime array virtual array can be constructed as Where l∈V\S means l belongs to V but not to S. The received data of the coprime virtual array can be expressed as X V (t) = [x V1 (t),x V2 (t),…,x Vmax(S) (t)] T , perform Hilbert transform on these data, and get the complex signal form of the virtual array receiving signal: Step 3: Use the complex signal of the virtual array Find the covariance matrix "(·) H ” is the conjugate transpose, where T is the number of snapshots in practical applications; Step 4. Select The non-zero columns in , get the new matrix where J = [e1,e2,...,e |S| ]∈R |V|×|S| is a selection matrix, e i is except the first (l i +1) positions except for the 1 column vector, Step 5. Generate matrix H ， H is a zero matrix of the same dimension as R. Set the elements in H corresponding to the zero entries in R to 0, and set the elements in the remaining positions to 1; Step 6: Fill the holes in R in step 4. The matrix filling problem can be implemented using the following atomic norm optimization model: By solving the corresponding semi-positive programming problem, R is optimized to fill the holes, using After optimization and in is a complex matrix, Is u∈C| V | ×1 is the Hermitian, Toeplitz, or positive semidefinite matrix of the first column vector, where Y∈C |S|×|S| "tr(·)" represents the trace of the matrix, "() H " represents the conjugate transpose of the matrix, min represents minimization, μ is the regularization parameter, represents the Hadamard product, "|| || F " represents the Frobenius norm of the matrix, "≥" represents semi-positive definite, and the semi-positive definite programming problem can be solved by the CVX convex optimization toolbox; Step 7: Divide into blocks and get where D∈C |V|×|K| , E∈C |V|×|V-K| , K is the number of incident signals; Step 8. Calculate D and E in step 7 to obtain The minimum solution is P=(D H D) -1 D H E; Step 9: The projection operator introduced into the noise subspace can be orthogonalized to obtain the matrix Q = [P H -I L-K ], where I represents the unit matrix, I∈R (L-K)×(L-K) , L is the length of the virtual array structure V; Step 10: According to the formula Calculate the spatial spectrum, where a(θ) represents the array steering vector of the incident signal. Perform a peak search on the spatial spectrum, and the angle corresponding to its extreme value is the estimated value of the incident signal angle.

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